# Competition between Variable–Supply and Fixed–Supply Currencies

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Background

#### 1.2. Contribution

## 2. Literature

#### 2.1. Currency Competition

#### 2.2. Competition between Fiat Currencies and Cryptocurrencies

#### 2.3. CBDCs and Cryptocurrencies

#### 2.4. The Cryptocurrency Market

#### 2.5. Game Theoretic Analyses and Decision Models

#### 2.6. Literature Summary and Additions to the Literature Gap

## 3. The Model

#### 3.1. One Variable–Supply Fiat Currency $n$

#### 3.2. One Variable–Supply Fiat Currency $n$ Competing with One Fixed–Supply Currency g

#### 3.3. Replicator Dynamics

## 4. Analyzing the Model

#### 4.1. The US 1635–2021

#### 4.2. Analysis Applying Replicator Dynamics

## 5. Summarizing the Results

## 6. Discussion, Policy Implications, Limitations, and Future Research

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

Parameters | |

$n$ | Variable–supply fiat currency |

$g$ | Fixed–supply currency |

${t}_{0}$ | Initial time, ${t}_{0}\ge 0$ |

$T$ | Final time, $T\ge {t}_{0}$ |

$i$ | Time counting variable, ${t}_{0}\le i\le T$ |

$\tau $ | Time lag from money printing to inflation, $\tau \ge 0$ |

${s}_{i}$ | Supply at discrete time $i$ of the variable–supply fiat currency $n$, ${s}_{i}\in \mathbb{R}$ |

${\pi}_{i}$ | Inflation at time $i$, ${\pi}_{i}\in \mathbb{R}$ |

$\alpha $ | Cobb–Douglas elasticity expressing weight assigned to money printing, $0\le \alpha \le 1$ |

${h}_{t}$ | The player’s support of currency $g$ relative to currency $n$ at time $t$, $0\le {h}_{t}\le 1$ |

$k$ | Parameter for the sensitivity or rapidity of change of the replicator equation, $k>0$ |

Independent variables | |

$t$ | Time, $t\ge {t}_{0}$ |

${p}_{nt}$ | Volume fraction of the player’s transactions in currency $n$ at time $t$, $0\le {p}_{t}\le 1$ |

Dependent variables | |

${u}_{nt}$ | Player’s Cobb–Douglas utility of holding currency $n$ at time $t$, ${u}_{nt}\ge 0$ |

${u}_{nMt}$ | Player’s utility of holding currency $n$ at time $t$ based on money printing, ${u}_{nMt}\ge 0$ |

${u}_{nIt}$ | Player’s utility of holding currency $n$ at time $t$ based on inflation, ${u}_{nIt}\ge 0$ |

${u}_{ngt}$ | Player’s utility of holding currency $n$ at time $t$ when currency $g$ is available,${u}_{ngt}\ge 0$ |

${u}_{gnt}$ | Player’s utility of holding currency $g$ at time $t$ when currency $n$ is available, ${u}_{gnt}\ge 0$ |

${u}_{t}$ | Player’s utility of holding currencies $n$ and $g$ at time $t$, ${u}_{t}\ge 0$ |

## Notes

1 | https://coinmarketcap.com/, retrieved 11 July 2022. |

2 | In total, 197,576 metric tons have been mined (gold.org 2022), and 3030 metric tons were produced in 2020 (Basov 2022). |

3 | We may operationalize ${h}_{t}$ as comprising six factors, i.e., backing (of currency $n$ relative to currency $g$) by actors, systems, or characteristics that users respect and trust; convenience, e.g., few and easily understood operations when purchasing goods and services; confidentiality, striking balances between privacy, availability, accessibility, and discrimination; transaction efficiency, i.e., low cost, fast speed, affordability, and finality in terms of how many confirmations are needed for transactional approval; financial stability, which usually depends on conditions in the given country; and security, see, e.g., Allen et al. (2020) and Kiff et al. (2020) for the security of blockchain–based currencies. |

4 | For the special case that $k\left({u}_{ngt}-{u}_{gnt}\right)=K{t}^{m}$ where $K$ and $m$ are parameters, which depend on time $t$ in a special manner and depend on time $t$ when $m=0$, the solution of (7) is ${p}_{t}=\frac{1}{1+\left(\frac{1}{{p}_{{t}_{0}}}-1\right){e}^{-\frac{K}{1+m}\left({t}^{1+m}-{\displaystyle {t}_{0}^{1+m}}\right)}}$, where $\frac{1}{{p}_{{t}_{0}}}-1>0$ when $0\le {p}_{{t}_{0}}<1$, $\underset{t\to \infty}{\mathrm{lim}}}{e}^{-\frac{K}{1+m}\left({t}^{1+m}-{\displaystyle {t}_{0}^{1+m}}\right)}=0$ causing $\underset{t\to \infty}{\mathrm{lim}}}{p}_{t}=1$ when $\frac{K}{1+m}>0$, $\underset{t\to \infty}{\mathrm{lim}}}{e}^{-\frac{K}{1+m}\left({t}^{1+m}-{\displaystyle {t}_{0}^{1+m}}\right)}=\infty $ causing $\underset{t\to \infty}{\mathrm{lim}}}{p}_{t}=0$ when $\frac{K}{1+m}<0$, and $\underset{t\to \infty}{\mathrm{lim}}}{p}_{t}={p}_{{t}_{0}$ when $\frac{K}{1+m}=0$. Hence, either one currency excludes the other currency, or the fraction ${p}_{t}$ equals the initial fraction ${p}_{{t}_{0}}$ at time ${t}_{0}$. |

5 | https://www.atlanticcouncil.org/cbdctracker/, retrieved 12 October 2022. |

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**Figure 1.**Panel (

**a**): US M2 money supply ${s}_{i}$ 1959–2021 in USD billion. Panel (

**b**): US inflation ${\pi}_{i}$ 1959–2021. Panels (

**c**–

**h**): The player’s utility ${u}_{nt},{u}_{nMt},{u}_{nIt}$ as a function of time $t$ for the Cobb–Douglas elasticities $\alpha =0.6,\text{}0.5,\text{}0.4,\text{}0.3,\text{}0.2$. Panel (

**c**): Equation (1) 1959–2021. Panel (

**d**): Equation (1) 1959–1975. Panel (

**e**): Equation (2) 1959–2021 based on money printing empirics. Panel (

**f**): Equation (3) 1959–2021 based on inflation empirics. Panel (

**g**): Equation (3) 1635–2021 based on inflation empirics. Panel (

**h**): Equation (3) 1695–2021 based on inflation empirics.

**Figure 2.**The fraction ${p}_{nt}$ of the player’s volume of transactions in currency $n$ at time $t$ 1959–2021 when $k={p}_{n{t}_{0}}=0.5$, applying the empirics in Figure 1c. Panels (

**a**,

**c**,

**e**): $\alpha =0.6$. Panels (

**b**,

**d**,

**f**): $\alpha =0.2$. Panels (

**a**,

**b**): Seven constant support parameters between ${h}_{t}=0.01$ and ${h}_{t}=0.99$. Panels (

**c**,

**d**): Seven linearly increasing support parameters ${h}_{t}$. Panels (

**e**,

**f**): Seven linearly decreasing support parameters ${h}_{t}$.

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## Share and Cite

**MDPI and ACS Style**

Wang, G.; Hausken, K.
Competition between Variable–Supply and Fixed–Supply Currencies. *Economies* **2022**, *10*, 270.
https://doi.org/10.3390/economies10110270

**AMA Style**

Wang G, Hausken K.
Competition between Variable–Supply and Fixed–Supply Currencies. *Economies*. 2022; 10(11):270.
https://doi.org/10.3390/economies10110270

**Chicago/Turabian Style**

Wang, Guizhou, and Kjell Hausken.
2022. "Competition between Variable–Supply and Fixed–Supply Currencies" *Economies* 10, no. 11: 270.
https://doi.org/10.3390/economies10110270